Copyright	(c) Marco Zocca 2017
License	GPL-3 (see the file LICENSE)
Maintainer	zocca marco gmail
Stability	experimental
Portability	portable
Safe Haskell	None
Language	Haskell2010

Numeric.LinearAlgebra.Class

Contents

Matrix and vector elements (optionally Complex)
Additive group
Vector space v.
- Hilbert-space distance function
Normed vector spaces
Matrix ring
Linear vector space
- LinearVectorSpace + Normed
- Linear systems
FiniteDim : finite-dimensional objects
HasData : accessing inner data (do not export)
Sparse : sparse datastructures
Set : types that behave as sets
SpContainer : sparse container datastructures. Insertion, lookup, toList, lookup with 0 default
SparseVector
SparseMatrix
SparseMatVec
Utilities

Description

Typeclasses for linear algebra and related concepts

Synopsis

class (Eq e, Fractional e, Floating e, Num (EltMag e), Ord (EltMag e)) => Elt e where
- type EltMag e :: *
class AdditiveGroup v where
class (AdditiveGroup v, Num (Scalar v)) => VectorSpace v where
- type Scalar v :: *
class VectorSpace v => InnerSpace v where
dot :: InnerSpace v => v -> v -> Scalar v
(./) :: (VectorSpace v, s ~ Scalar v, Fractional s) => v -> s -> v
(*.) :: (VectorSpace v, s ~ Scalar v) => v -> s -> v
cvx :: VectorSpace v => Scalar v -> v -> v -> v
hilbertDistSq :: InnerSpace v => v -> v -> Scalar v
class (InnerSpace v, Num (RealScalar v), Eq (RealScalar v), Epsilon (Magnitude v), Show (Magnitude v), Ord (Magnitude v)) => Normed v where
- type Magnitude v :: *
- type RealScalar v :: *
normInftyR :: (Foldable t, Ord a) => t a -> a
normInftyC :: (Foldable t, RealFloat a, Functor t) => t (Complex a) -> a
dotLp :: (Set t, Foldable t, Floating a) => a -> t a -> t a -> a
reciprocal :: (Functor f, Fractional b) => f b -> f b
scale :: (Num b, Functor f) => b -> f b -> f b
class (AdditiveGroup m, Epsilon (MatrixNorm m)) => MatrixRing m where
- type MatrixNorm m :: *
class VectorSpace v => LinearVectorSpace v where
- type MatrixType v :: *
type V v = (LinearVectorSpace v, Normed v)
class LinearVectorSpace v => LinearSystem v where
class FiniteDim f where
- type FDSize f
class HasData f where
- type HDData f
class (FiniteDim f, HasData f) => Sparse f where
class Functor f => Set f where
class Sparse c => SpContainer c where
- type ScIx c :: *
- type ScElem c
class SpContainer v => SparseVector v where
- type SpvIx v :: *
class SpContainer m => SparseMatrix m where
toC :: Num a => a -> Complex a

Matrix and vector elements (optionally Complex)

class (Eq e, Fractional e, Floating e, Num (EltMag e), Ord (EltMag e)) => Elt e where #

Minimal complete definition

mag

Associated Types

type EltMag e :: * #

Methods

conj :: e -> e #

Complex conjugate, or identity function if its input is real-valued

mag :: e -> EltMag e #

Magnitude

Instances

Elt Double #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type EltMag Double :: * # Methods conj :: Double -> Double # mag :: Double -> EltMag Double #
Elt Float #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type EltMag Float :: * # Methods conj :: Float -> Float # mag :: Float -> EltMag Float #
RealFloat e => Elt (Complex e) #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type EltMag (Complex e) :: * # Methods conj :: Complex e -> Complex e # mag :: Complex e -> EltMag (Complex e) #

Additive group

class AdditiveGroup v where #

Minimal complete definition

zeroV, (^+^), negateV

Methods

zeroV :: v #

The zero element: identity for '(^+^)'

(^+^) :: v -> v -> v infixl 6 #

Add vectors

negateV :: v -> v #

Additive inverse

(^-^) :: v -> v -> v infixl 6 #

Group subtraction

Instances

AdditiveGroup Double #
Instance details Defined in Numeric.LinearAlgebra.Class Methods zeroV :: Double # (^+^) :: Double -> Double -> Double # negateV :: Double -> Double # (^-^) :: Double -> Double -> Double #
AdditiveGroup Float #	Instances for builtin types
Instance details Defined in Numeric.LinearAlgebra.Class Methods zeroV :: Float # (^+^) :: Float -> Float -> Float # negateV :: Float -> Float # (^-^) :: Float -> Float -> Float #
AdditiveGroup (Complex Double) #
Instance details Defined in Numeric.LinearAlgebra.Class Methods zeroV :: Complex Double # (^+^) :: Complex Double -> Complex Double -> Complex Double # negateV :: Complex Double -> Complex Double # (^-^) :: Complex Double -> Complex Double -> Complex Double #
AdditiveGroup (Complex Float) #
Instance details Defined in Numeric.LinearAlgebra.Class Methods zeroV :: Complex Float # (^+^) :: Complex Float -> Complex Float -> Complex Float # negateV :: Complex Float -> Complex Float # (^-^) :: Complex Float -> Complex Float -> Complex Float #
AdditiveGroup a => AdditiveGroup (SpVector a) #
Instance details Defined in Data.Sparse.SpVector Methods zeroV :: SpVector a # (^+^) :: SpVector a -> SpVector a -> SpVector a # negateV :: SpVector a -> SpVector a # (^-^) :: SpVector a -> SpVector a -> SpVector a #
AdditiveGroup a => AdditiveGroup (SpMatrix a) #	`SpMatrix`es form an additive group, in that they can have an invertible associtative operation (matrix sum)
Instance details Defined in Data.Sparse.SpMatrix Methods zeroV :: SpMatrix a # (^+^) :: SpMatrix a -> SpMatrix a -> SpMatrix a # negateV :: SpMatrix a -> SpMatrix a # (^-^) :: SpMatrix a -> SpMatrix a -> SpMatrix a #

Vector space `v`.

class (AdditiveGroup v, Num (Scalar v)) => VectorSpace v where #

Minimal complete definition

(.*)

Associated Types

type Scalar v :: * #

Methods

(.*) :: Scalar v -> v -> v infixr 7 #

Scale a vector

Instances

VectorSpace Double #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type Scalar Double :: * # Methods (.*) :: Scalar Double -> Double -> Double #
VectorSpace Float #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type Scalar Float :: * # Methods (.*) :: Scalar Float -> Float -> Float #
VectorSpace (Complex Double) #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type Scalar (Complex Double) :: * # Methods (.*) :: Scalar (Complex Double) -> Complex Double -> Complex Double #
VectorSpace (Complex Float) #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type Scalar (Complex Float) :: * # Methods (.*) :: Scalar (Complex Float) -> Complex Float -> Complex Float #
VectorSpace a => VectorSpace (SpVector a) #
Instance details Defined in Data.Sparse.SpVector Associated Types type Scalar (SpVector a) :: * # Methods (.*) :: Scalar (SpVector a) -> SpVector a -> SpVector a #
VectorSpace a => VectorSpace (SpMatrix a) #
Instance details Defined in Data.Sparse.SpMatrix Associated Types type Scalar (SpMatrix a) :: * # Methods (.*) :: Scalar (SpMatrix a) -> SpMatrix a -> SpMatrix a #

class VectorSpace v => InnerSpace v where #

Adds inner (dot) products.

Minimal complete definition

(<.>)

Methods

(<.>) :: v -> v -> Scalar v #

Inner/dot product

Instances

InnerSpace Double #
Instance details Defined in Numeric.LinearAlgebra.Class Methods (<.>) :: Double -> Double -> Scalar Double #
InnerSpace Float #
Instance details Defined in Numeric.LinearAlgebra.Class Methods (<.>) :: Float -> Float -> Scalar Float #
InnerSpace (Complex Double) #
Instance details Defined in Numeric.LinearAlgebra.Class Methods (<.>) :: Complex Double -> Complex Double -> Scalar (Complex Double) #
InnerSpace (Complex Float) #
Instance details Defined in Numeric.LinearAlgebra.Class Methods (<.>) :: Complex Float -> Complex Float -> Scalar (Complex Float) #
InnerSpace a => InnerSpace (SpVector a) #
Instance details Defined in Data.Sparse.SpVector Methods (<.>) :: SpVector a -> SpVector a -> Scalar (SpVector a) #

dot :: InnerSpace v => v -> v -> Scalar v #

Inner product

(./) :: (VectorSpace v, s ~ Scalar v, Fractional s) => v -> s -> v infixr 7 #

Scale a vector by the reciprocal of a number (e.g. for normalization)

(*.) :: (VectorSpace v, s ~ Scalar v) => v -> s -> v infixl 7 #

Vector multiplied by scalar

cvx :: VectorSpace v => Scalar v -> v -> v -> v #

Convex combination of two vectors (NB: 0 <= a <= 1).

Hilbert-space distance function

hilbertDistSq :: InnerSpace v => v -> v -> Scalar v #

`hilbertDistSq x y = || x - y ||^2` computes the squared L2 distance between two vectors

Normed vector spaces

class (InnerSpace v, Num (RealScalar v), Eq (RealScalar v), Epsilon (Magnitude v), Show (Magnitude v), Ord (Magnitude v)) => Normed v where #

Minimal complete definition

norm1, norm2Sq, normP, normalize, normalize2

Associated Types

type Magnitude v :: * #

type RealScalar v :: * #

Methods

norm1 :: v -> Magnitude v #

L1 norm

norm2Sq :: v -> Magnitude v #

Euclidean (L2) norm squared

normP :: RealScalar v -> v -> Magnitude v #

Lp norm (p > 0)

normalize :: RealScalar v -> v -> v #

Normalize w.r.t. Lp norm

normalize2 :: v -> v #

Normalize w.r.t. L2 norm

normalize2' :: Floating (Scalar v) => v -> v #

Normalize w.r.t. norm2' instead of norm2

norm2 :: Floating (Magnitude v) => v -> Magnitude v #

Euclidean (L2) norm

norm2' :: Floating (Scalar v) => v -> Scalar v #

Euclidean (L2) norm; returns a Complex (norm :+ 0) for Complex-valued vectors

norm :: Floating (Magnitude v) => RealScalar v -> v -> Magnitude v #

Lp norm (p > 0)

Instances

Normed Double #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type Magnitude Double :: * # type RealScalar Double :: * # Methods norm1 :: Double -> Magnitude Double # norm2Sq :: Double -> Magnitude Double # normP :: RealScalar Double -> Double -> Magnitude Double # normalize :: RealScalar Double -> Double -> Double # normalize2 :: Double -> Double # normalize2' :: Double -> Double # norm2 :: Double -> Magnitude Double # norm2' :: Double -> Scalar Double # norm :: RealScalar Double -> Double -> Magnitude Double #
Normed Float #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type Magnitude Float :: * # type RealScalar Float :: * # Methods norm1 :: Float -> Magnitude Float # norm2Sq :: Float -> Magnitude Float # normP :: RealScalar Float -> Float -> Magnitude Float # normalize :: RealScalar Float -> Float -> Float # normalize2 :: Float -> Float # normalize2' :: Float -> Float # norm2 :: Float -> Magnitude Float # norm2' :: Float -> Scalar Float # norm :: RealScalar Float -> Float -> Magnitude Float #
Normed (Complex Double) #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type Magnitude (Complex Double) :: * # type RealScalar (Complex Double) :: * # Methods norm1 :: Complex Double -> Magnitude (Complex Double) # norm2Sq :: Complex Double -> Magnitude (Complex Double) # normP :: RealScalar (Complex Double) -> Complex Double -> Magnitude (Complex Double) # normalize :: RealScalar (Complex Double) -> Complex Double -> Complex Double # normalize2 :: Complex Double -> Complex Double # normalize2' :: Complex Double -> Complex Double # norm2 :: Complex Double -> Magnitude (Complex Double) # norm2' :: Complex Double -> Scalar (Complex Double) # norm :: RealScalar (Complex Double) -> Complex Double -> Magnitude (Complex Double) #
Normed (Complex Float) #
Instance details Defined in Numeric.LinearAlgebra.Class Associated Types type Magnitude (Complex Float) :: * # type RealScalar (Complex Float) :: * # Methods norm1 :: Complex Float -> Magnitude (Complex Float) # norm2Sq :: Complex Float -> Magnitude (Complex Float) # normP :: RealScalar (Complex Float) -> Complex Float -> Magnitude (Complex Float) # normalize :: RealScalar (Complex Float) -> Complex Float -> Complex Float # normalize2 :: Complex Float -> Complex Float # normalize2' :: Complex Float -> Complex Float # norm2 :: Complex Float -> Magnitude (Complex Float) # norm2' :: Complex Float -> Scalar (Complex Float) # norm :: RealScalar (Complex Float) -> Complex Float -> Magnitude (Complex Float) #
(Normed a, Magnitude a ~ RealScalar a, RealScalar a ~ Scalar a) => Normed (SpVector a) #
Instance details Defined in Data.Sparse.SpVector Associated Types type Magnitude (SpVector a) :: * # type RealScalar (SpVector a) :: * # Methods norm1 :: SpVector a -> Magnitude (SpVector a) # norm2Sq :: SpVector a -> Magnitude (SpVector a) # normP :: RealScalar (SpVector a) -> SpVector a -> Magnitude (SpVector a) # normalize :: RealScalar (SpVector a) -> SpVector a -> SpVector a # normalize2 :: SpVector a -> SpVector a # normalize2' :: SpVector a -> SpVector a # norm2 :: SpVector a -> Magnitude (SpVector a) # norm2' :: SpVector a -> Scalar (SpVector a) # norm :: RealScalar (SpVector a) -> SpVector a -> Magnitude (SpVector a) #

normInftyR :: (Foldable t, Ord a) => t a -> a #

Infinity-norm (Real)

normInftyC :: (Foldable t, RealFloat a, Functor t) => t (Complex a) -> a #

Infinity-norm (Complex)

dotLp :: (Set t, Foldable t, Floating a) => a -> t a -> t a -> a #

Lp inner product (p > 0)

reciprocal :: (Functor f, Fractional b) => f b -> f b #

Reciprocal

scale :: (Num b, Functor f) => b -> f b -> f b #

Scale

Matrix ring

class (AdditiveGroup m, Epsilon (MatrixNorm m)) => MatrixRing m where #

A matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication

Minimal complete definition

(##), (##^), transpose, normFrobenius

Associated Types

type MatrixNorm m :: * #

Methods

(##) :: m -> m -> m #

Matrix-matrix product

(##^) :: m -> m -> m #

Matrix times matrix transpose (A B^T)

(#^#) :: m -> m -> m #

Matrix transpose times matrix (A^T B)

transpose :: m -> m #

Matrix transpose (Hermitian conjugate in the Complex case)

normFrobenius :: m -> MatrixNorm m #

Frobenius norm

Instances

MatrixRing (SpMatrix Double) #
Instance details Defined in Data.Sparse.SpMatrix Associated Types type MatrixNorm (SpMatrix Double) :: * # Methods (##) :: SpMatrix Double -> SpMatrix Double -> SpMatrix Double # (##^) :: SpMatrix Double -> SpMatrix Double -> SpMatrix Double # (#^#) :: SpMatrix Double -> SpMatrix Double -> SpMatrix Double # transpose :: SpMatrix Double -> SpMatrix Double # normFrobenius :: SpMatrix Double -> MatrixNorm (SpMatrix Double) #
MatrixRing (SpMatrix (Complex Double)) #
Instance details Defined in Data.Sparse.SpMatrix Associated Types type MatrixNorm (SpMatrix (Complex Double)) :: * # Methods (##) :: SpMatrix (Complex Double) -> SpMatrix (Complex Double) -> SpMatrix (Complex Double) # (##^) :: SpMatrix (Complex Double) -> SpMatrix (Complex Double) -> SpMatrix (Complex Double) # (#^#) :: SpMatrix (Complex Double) -> SpMatrix (Complex Double) -> SpMatrix (Complex Double) # transpose :: SpMatrix (Complex Double) -> SpMatrix (Complex Double) # normFrobenius :: SpMatrix (Complex Double) -> MatrixNorm (SpMatrix (Complex Double)) #

Linear vector space

class VectorSpace v => LinearVectorSpace v where #

Minimal complete definition

(#>), (<#)

Associated Types

type MatrixType v :: * #

Methods

(#>) :: MatrixType v -> v -> v #

Matrix-vector action

(<#) :: v -> MatrixType v -> v #

Dual matrix-vector action

Instances

(InnerSpace t, Scalar t ~ t) => LinearVectorSpace (SpVector t) #
Instance details Defined in Data.Sparse.Common Associated Types type MatrixType (SpVector t) :: * # Methods (#>) :: MatrixType (SpVector t) -> SpVector t -> SpVector t # (<#) :: SpVector t -> MatrixType (SpVector t) -> SpVector t #

LinearVectorSpace + Normed

type V v = (LinearVectorSpace v, Normed v) #

Linear systems

class LinearVectorSpace v => LinearSystem v where #

Minimal complete definition

(<\>)

Methods

(<\>) #

Arguments

:: (MonadIO m, MonadThrow m)
=> MatrixType v	System matrix
-> v	Right-hand side
-> m v	Result

Solve a linear system; uses GMRES internally as default method

Instances

LinearSystem (SpVector Double) #	<> uses the GMRES method as default
Instance details Defined in Numeric.LinearAlgebra.Sparse Methods (<\>) :: (MonadIO m, MonadThrow m) => MatrixType (SpVector Double) -> SpVector Double -> m (SpVector Double) #

FiniteDim : finite-dimensional objects

class FiniteDim f where #

Minimal complete definition

dim

Associated Types

type FDSize f #

Methods

dim :: f -> FDSize f #

Dimension (i.e. Int for SpVector, (Int, Int) for SpMatrix)

Instances

FiniteDim (SpVector a) #	`SpVector`s form a vector space because they can be multiplied by a scalar `SpVector`s are finite-dimensional vectors
Instance details Defined in Data.Sparse.SpVector Associated Types type FDSize (SpVector a) :: * # Methods dim :: SpVector a -> FDSize (SpVector a) #
FiniteDim (SpMatrix a) #	`SpMatrix`es are maps between finite-dimensional spaces
Instance details Defined in Data.Sparse.SpMatrix Associated Types type FDSize (SpMatrix a) :: * # Methods dim :: SpMatrix a -> FDSize (SpMatrix a) #

HasData : accessing inner data (do not export)

class HasData f where #

Minimal complete definition

nnz, dat

Associated Types

type HDData f #

Methods

nnz :: f -> Int #

Number of nonzeros

dat :: f -> HDData f #

Instances

HasData (SpVector a) #
Instance details Defined in Data.Sparse.SpVector Associated Types type HDData (SpVector a) :: * # Methods nnz :: SpVector a -> Int # dat :: SpVector a -> HDData (SpVector a) #
HasData (SpMatrix a) #
Instance details Defined in Data.Sparse.SpMatrix Associated Types type HDData (SpMatrix a) :: * # Methods nnz :: SpMatrix a -> Int # dat :: SpMatrix a -> HDData (SpMatrix a) #

Sparse : sparse datastructures

class (FiniteDim f, HasData f) => Sparse f where #

Minimal complete definition

spy

Methods

spy :: Fractional b => f -> b #

Sparsity (fraction of nonzero elements)

Instances

Sparse (SpVector a) #
Instance details Defined in Data.Sparse.SpVector Methods spy :: Fractional b => SpVector a -> b #
Sparse (SpMatrix a) #
Instance details Defined in Data.Sparse.SpMatrix Methods spy :: Fractional b => SpMatrix a -> b #

Set : types that behave as sets

class Functor f => Set f where #

Minimal complete definition

liftU2, liftI2

Methods

liftU2 :: (a -> a -> a) -> f a -> f a -> f a #

Union binary lift : apply function on _union_ of two "sets"

liftI2 :: (a -> a -> b) -> f a -> f a -> f b #

Intersection binary lift : apply function on _intersection_ of two "sets"

Instances

Set SpVector #
Instance details Defined in Data.Sparse.SpVector Methods liftU2 :: (a -> a -> a) -> SpVector a -> SpVector a -> SpVector a # liftI2 :: (a -> a -> b) -> SpVector a -> SpVector a -> SpVector b #
Set SpMatrix #
Instance details Defined in Data.Sparse.SpMatrix Methods liftU2 :: (a -> a -> a) -> SpMatrix a -> SpMatrix a -> SpMatrix a # liftI2 :: (a -> a -> b) -> SpMatrix a -> SpMatrix a -> SpMatrix b #

SpContainer : sparse container datastructures. Insertion, lookup, toList, lookup with 0 default

class Sparse c => SpContainer c where #

Minimal complete definition

scInsert, scLookup, scToList, (@@)

Associated Types

type ScIx c :: * #

type ScElem c #

Methods

scInsert :: ScIx c -> ScElem c -> c -> c #

scLookup :: c -> ScIx c -> Maybe (ScElem c) #

scToList :: c -> [(ScIx c, ScElem c)] #

(@@) :: c -> ScIx c -> ScElem c #

Instances

Elt a => SpContainer (SpVector a) #	`SpVector`s are sparse containers too, i.e. any specific component may be missing (so it is assumed to be 0)
Instance details Defined in Data.Sparse.SpVector Associated Types type ScIx (SpVector a) :: * # type ScElem (SpVector a) :: * # Methods scInsert :: ScIx (SpVector a) -> ScElem (SpVector a) -> SpVector a -> SpVector a # scLookup :: SpVector a -> ScIx (SpVector a) -> Maybe (ScElem (SpVector a)) # scToList :: SpVector a -> [(ScIx (SpVector a), ScElem (SpVector a))] # (@@) :: SpVector a -> ScIx (SpVector a) -> ScElem (SpVector a) #
Num a => SpContainer (SpMatrix a) #	`SpMatrix`es are sparse containers too, i.e. any specific component may be missing (so it is assumed to be 0)
Instance details Defined in Data.Sparse.SpMatrix Associated Types type ScIx (SpMatrix a) :: * # type ScElem (SpMatrix a) :: * # Methods scInsert :: ScIx (SpMatrix a) -> ScElem (SpMatrix a) -> SpMatrix a -> SpMatrix a # scLookup :: SpMatrix a -> ScIx (SpMatrix a) -> Maybe (ScElem (SpMatrix a)) # scToList :: SpMatrix a -> [(ScIx (SpMatrix a), ScElem (SpMatrix a))] # (@@) :: SpMatrix a -> ScIx (SpMatrix a) -> ScElem (SpMatrix a) #